Aim: How do we evaluate and solve problems involving surface area of prisms an cylinders?
Objective: Learn and apply the formula for the surface area of a prism. Learn and apply the formula for the surface area of a cylinder.
Vocabulary: lateral face, lateral edge, right prism, oblique prism, altitude, surface area, lateral surface, axis of a cylinder, right cylinder, oblique cylinder.
Prisms and cylinders have 2 congruent parallel bases.
A lateral face is not a base. The edges of the base are called base edges. A lateral edge is not an edge of a base. The lateral faces of a right prism are all rectangles. An oblique prism has at least one nonrectangular lateral face.
An altitude of a prism or cylinder is a perpendicular segment joining the planes of the bases. The height of a three-dimensional figure is the length of an altitude.
Surface area is the total area of all faces and curved surfaces of a three-dimensional figure. The lateral area of a prism is the sum of the areas of the lateral faces.
Holt Geometry
The net of a right prism can be drawn so that the lateral faces form a rectangle with the same height as the prism. The base of the rectangle is equal to the perimeter of the base of the prism.
The surface area of a right rectangular prism with length ℓ, width w, and height h can be written as
S = 2ℓw + 2wh + 2ℓh.
The surface area formula is only true for right prisms. To find the surface area of an oblique prism, add the areas of the faces.
Example 1A: Finding Lateral Areas and Surface Areas of Prisms
Find the lateral area and surface area of the right rectangular prism. Round to the nearest tenth, if necessary.
This is one method.
L .A.= Ph P = 2(9) + 2(7) = 32 ft
L.A = 32(14) = 448 ft2
S = Ph + 2B
= 448 + 2(7)(9) = 574 ft2
Holt Geometry
Another Method is find the area of each face. We know that opposite faces are congruent.
L.A. = Area of 2 [(7) x ( 14) ] + 2 [ (9) x (14)]
This is the area of one This is the area of the front side. We multiplied by side. We multiplied by two two because the because the opposite side opposite side is is congruent. congruent.
L .A. = 2 [98] + 2 [126]
L.A. = 196 + 252
2 L.A. = 448ft
Surface area is lateral area plus the area of the base:
448 + 2 [(7)(9)]
Lateral Area This is the area of the bases. Which is the top and the bottom of the pyramid.
S.A. = 448 + 126
2 S.A = 574ft
Both methods you will get the same answer.
Holt Geometry
The lateral surface of a cylinder is the curved surface that connects the two bases. The axis of a cylinder is the segment with endpoints at the centers of the bases. The axis of a right cylinder is perpendicular to its bases. The axis of an oblique cylinder is not perpendicular to its bases. The altitude of a right cylinder is the same length as the axis.
Example 2A: Finding Lateral Areas and Surface Areas of Right Cylinders
Find the lateral area and surface area of the right cylinder. Give your answers in terms of .
The radius is half the diameter, or 8 ft.
L = 2rh = 2(8)(10) = 160 in2
S = L + 2r2 = 160 + 2(8)2
= 288 in2
Do these Problem:
1. Find the lateral area and surface area of a right cylinder with circumference 24 cm and a height equal to half the radius. Give your answers in terms of . 2. Find the lateral area and surface area of a cylinder with a base area of 49 and a height that is 2 times the radius.
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3. Find the surface area of the composite figure
4. The edge length of the cube is tripled. Describe the effect on the surface area.
5. Find the lateral area and the surface area of each figure. Round to the nearest tenth, if necessary.
a. A cube with edge length 10 cm
b. A regular hexagonal prism with height 15 in. and base edge length 8 in.
Holt Geometry